Ranking Specific Sets of Objects
|
|
|
- Gwendoline Eaton
- 6 years ago
- Views:
Transcription
1 Ranking Specific Sets of Objects Jan Maly, Stefan Woltran BTW 2017, Stuttgart March 7, 2017
2 Lifting rankings from objects to sets Given A set S A linear order < on S A family X P(S) \ { } of nonempty subsets of S The problem Is there a good ranking on X? Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 1
3 An example - Basics S = {strawberry, vanilla, chocolate, lemon} Linear order: strawberry < chocolate < vanilla < lemon X = {{strawberry}, {chocolate}, {strawberry, vanilla}, {strawberry, vanilla, lemon}, {strawberry, lemon}} Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 2
4 The axiomatic approach What is a good ranking? The ranking should be based on the linear order < The ranking should be transitive, either reflexive or irreflexive,... good depends on the interpretation of X Possible interpretations Sets as final outcomes Opportunities Complete uncertainty etc... Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 3
5 Axioms for ranking sets under complete uncertainty Extension Rule For all x, y S if {x}, {y} X, then Dominance {x} {y} iff x < y For all A X and all x S if A {x} X, then y < x for all y A implies A A {x} x < y for all y A implies A {x} A If X = P(S) \ { } and is transitive, the extension rule is implied by dominance. Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 4
6 An example - extension rule and dominance S = {strawberry, vanilla, chocolate, lemon} Linear order: strawberry < chocolate < vanilla < lemon X = {{strawberry}, {chocolate}, {strawberry, vanilla}, {strawberry, vanilla, lemon}, {strawberry, lemon}} Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 5
7 An example - extension rule and dominance S = {strawberry, vanilla, chocolate, lemon} Linear order: strawberry < chocolate < vanilla < lemon X = {{strawberry}, {chocolate}, {strawberry, vanilla}, {strawberry, vanilla, lemon}, {strawberry, lemon}} Extension rule: {strawberry} {chocolate} Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 5
8 An example - extension rule and dominance S = {strawberry, vanilla, chocolate, lemon} Linear order: strawberry < chocolate < vanilla < lemon X = {{strawberry}, {chocolate}, {strawberry, vanilla}, {strawberry, vanilla, lemon}, {strawberry, lemon}} Extension rule: {strawberry} {chocolate} Dominance: {strawberry} {strawberry, vanilla} {strawberry, vanilla, lemon} {strawberry} {strawberry, lemon} Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 5
9 Axioms for ranking sets under complete uncertainty Independence For all A, B X and for all x S \ (A B) if A {x}, B {x} X, then A B implies A {x} B {x} Strict Independence For all A, B X and for all x S \ (A B) if A {x}, B {x} X, then A B implies A {x} B {x} Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 6
10 An example - all axioms S = {strawberry, vanilla, chocolate, lemon} Linear order: strawberry < chocolate < vanilla < lemon X = {{strawberry}, {chocolate}, {strawberry, vanilla}, {strawberry, vanilla, lemon}, {strawberry, lemon}} Extension rule: {strawberry} {chocolate} Dominance: {strawberry} {strawberry, vanilla} {strawberry, vanilla, lemon} {strawberry} {strawberry, lemon} Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 7
11 An example - all axioms S = {strawberry, vanilla, chocolate, lemon} Linear order: strawberry < chocolate < vanilla < lemon X = {{strawberry}, {chocolate}, {strawberry, vanilla}, {strawberry, vanilla, lemon}, {strawberry, lemon}} Extension rule: {strawberry} {chocolate} Dominance: {strawberry} {strawberry, vanilla} {strawberry, vanilla, lemon} {strawberry} {strawberry, lemon} Independence: {strawberry, lemon} {strawberry, vanilla, lemon} Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 7
12 An example - all axioms S = {strawberry, vanilla, chocolate, lemon} Linear order: strawberry < chocolate < vanilla < lemon X = {{strawberry}, {chocolate}, {strawberry, vanilla}, {strawberry, vanilla, lemon}, {strawberry, lemon}} Extension rule: {strawberry} {chocolate} Dominance: {strawberry} {strawberry, vanilla} {strawberry, vanilla, lemon} {strawberry} {strawberry, lemon} Independence: {strawberry, lemon} {strawberry, vanilla, lemon} Strict independence: {strawberry, lemon} {strawberry, vanilla, lemon} Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 7
13 Classic impossibility results Kannai and Peleg (1984) Assume X = P(S) \ { } and S 6, then there exists no order on X satisfying dominance and independence. Barberà and Pattanaik (1984) Assume X = P(S) \ { } and S 3, then there exists no binary relation on X satisfying dominance and strict independence. Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 8
14 Proof of Barberà and Pattanaik Assume S = {1, 2, 3} (1) {1} {1, 2} and (2) {2, 3} {3} by dominance {1, 3} {1, 2, 3} by (1) and strict independence {1, 2, 3} {1, 3} by (2) and strict independence Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 9
15 Proof of Kannai and Peleg Observation: dominance and independence imply A {max(a), min(a)} Assume S = {1, 2,..., 6} {3}?{2, 5} Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 10
16 Proof of Kannai and Peleg Observation: dominance and independence imply A {max(a), min(a)} Assume S = {1, 2,..., 6} {3} {2, 5} Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 10
17 Proof of Kannai and Peleg Observation: dominance and independence imply A {max(a), min(a)} Assume S = {1, 2,..., 6} {3} {2, 5} {3, 6} {2, 5, 6} by independence Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 10
18 Proof of Kannai and Peleg Observation: dominance and independence imply A {max(a), min(a)} Assume S = {1, 2,..., 6} {3} {2, 5} {3, 6} {2, 5, 6} by independence {3, 4, 5, 6} {2, 3, 4, 5, 6} by observation Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 10
19 Proof of Kannai and Peleg Observation: dominance and independence imply A {max(a), min(a)} Assume S = {1, 2,..., 6} {3} {2, 5} {3, 6} {2, 5, 6} by independence {3, 4, 5, 6} {2, 3, 4, 5, 6} by observation This contradicts dominance! Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 10
20 Proof of Kannai and Peleg Observation: dominance and independence imply A {max(a), min(a)} Assume S = {1, 2,..., 6} {2, 5} {3} Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 10
21 Proof of Kannai and Peleg Observation: dominance and independence imply A {max(a), min(a)} Assume S = {1, 2,..., 6} {2, 5} {3} {4} Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 10
22 Proof of Kannai and Peleg Observation: dominance and independence imply A {max(a), min(a)} Assume S = {1, 2,..., 6} {2, 5} {4} Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 10
23 Proof of Kannai and Peleg Observation: dominance and independence imply A {max(a), min(a)} Assume S = {1, 2,..., 6} {2, 5} {4} {1, 4} {1, 2, 5} by independence Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 10
24 Proof of Kannai and Peleg Observation: dominance and independence imply A {max(a), min(a)} Assume S = {1, 2,..., 6} {2, 5} {4} {1, 4} {1, 2, 5} by independence {1, 2, 3, 4} {1, 2, 3, 4, 5} by observation Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 10
25 Proof of Kannai and Peleg Observation: dominance and independence imply A {max(a), min(a)} Assume S = {1, 2,..., 6} {2, 5} {4} {1, 4} {1, 2, 5} by independence {1, 2, 3, 4} {1, 2, 3, 4, 5} by observation This contradicts dominance! Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 10
26 Ditching the assumption X = P(S) \ { } In many applications X is subject to constraints. There are families X = P(S) \ { } with S > 6 such that there is an order on X satisfying dominance and (strict) independence. It can be argued that dominance is too weak in the general case. Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 11
27 An example - all axioms S = {strawberry, vanilla, chocolate, lemon} Linear order: strawberry < chocolate < vanilla < lemon X = {{strawberry}, {chocolate}, {strawberry, vanilla}, {strawberry, vanilla, lemon}, {strawberry, lemon}} Extension rule: {strawberry} {chocolate} Dominance: {strawberry} {strawberry, vanilla} {strawberry, vanilla, lemon} {strawberry} {strawberry, lemon} Independence: {strawberry, lemon} {strawberry, vanilla, lemon} Strict independence: {strawberry, lemon} {strawberry, vanilla, lemon} Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 12
28 An example - all axioms S = {strawberry, vanilla, chocolate, lemon} Linear order: strawberry < chocolate < vanilla < lemon X = {{strawberry}, {chocolate}, {strawberry, vanilla}, {strawberry, vanilla, lemon}, {strawberry, lemon}} Extension rule: {strawberry} {chocolate} Dominance: {strawberry} {strawberry, vanilla} {strawberry, vanilla, lemon} {strawberry} {strawberry, lemon} Independence: {strawberry, lemon} {strawberry, vanilla, lemon} Strict independence: {strawberry, lemon} {strawberry, vanilla, lemon} {strawberry, vanilla}? {strawberry, lemon} Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 12
29 A strengthening of dominance Solution: Define a stronger version of dominance Maximal dominance For all A, B X, (max(a) max(b) min(a) < min(b)) (max(a) < max(b) min(a) min(b)) A B Assuming X = P(S) \ { }, maximal dominance is implied by dominance and (strict) independence. Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 13
30 An example - maximal dominance S = {strawberry, vanilla, chocolate, lemon} Linear order: strawberry < chocolate < vanilla < lemon X = {{strawberry}, {chocolate}, {strawberry, vanilla}, {strawberry, vanilla, lemon}, {strawberry, lemon}} Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 14
31 An example - maximal dominance S = {strawberry, vanilla, chocolate, lemon} Linear order: strawberry < chocolate < vanilla < lemon X = {{strawberry}, {chocolate}, {strawberry, vanilla}, {strawberry, vanilla, lemon}, {strawberry, lemon}} Maximal dominance: {strawberry} {strawberry, vanilla} {strawberry, vanilla, lemon} {strawberry} {strawberry, lemon} Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 14
32 An example - maximal dominance S = {strawberry, vanilla, chocolate, lemon} Linear order: strawberry < chocolate < vanilla < lemon X = {{strawberry}, {chocolate}, {strawberry, vanilla}, {strawberry, vanilla, lemon}, {strawberry, lemon}} Maximal dominance: {strawberry} {strawberry, vanilla} {strawberry, vanilla, lemon} {strawberry} {strawberry, lemon} {strawberry} {chocolate} Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 14
33 An example - maximal dominance S = {strawberry, vanilla, chocolate, lemon} Linear order: strawberry < chocolate < vanilla < lemon X = {{strawberry}, {chocolate}, {strawberry, vanilla}, {strawberry, vanilla, lemon}, {strawberry, lemon}} Maximal dominance: {strawberry} {strawberry, vanilla} {strawberry, vanilla, lemon} {strawberry} {strawberry, lemon} {strawberry} {chocolate} {strawberry, vanilla} {strawberry, lemon} Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 14
34 The problems we treated The Partial (Max)-Dominance-(Strict)-Independence Problem Given a linearly ordered set S and a set X P(S) \ { }, decide if there is a partial order/preorder on X satisfying (maximal) dominance and (strict) independence. The (Max)-Dominance-(Strict)-Independence Problem Given a linearly ordered set S and a set X P(S) \ { }, decide if there is a (strict) total order on X satisfying (maximal) dominance and (strict) independence. Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 15
35 The Partial (Max)-Dominance-(Strict)-Independence Problem Theorem The Partial (Max)-Dominance-Independence Problem is trivial. We can define a preorder satisfying maximal dominance and independence. Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 16
36 The Partial (Max)-Dominance-(Strict)-Independence Problem Theorem The Partial (Max)-Dominance-Independence Problem is trivial. We can define a preorder satisfying maximal dominance and independence. Theorem The Partial (Max)-Dominance-Strict-Independence Problem is P-complete. We construct the minimal transitive relation satisfying (maximal) dominance and strict independence, then check if this relation is irreflexive. We can prove the P-hardness by a reduction from Horn-Sat. Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 16
37 The Total (Max)-Dominance-(Strict)-Independence Problem The (Max)-Dominance-(Strict)-Independence problem is NP-hard. This can be shown via a reduction from betweenness. The Betweenness Problem Given a finite set V = {v 1, v 2,..., v n } and a set of triples R V 3, find a strict total order on V such that a < b < c or a > b > c holds for all (a, b, c) R. The NP-hardness of betweenness was shown 1979 by Jaroslav Opatrny. Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 17
38 The Total Max-Dominance-Strict-Independence Problem Idea: Represent the elements v 1, v 2,..., v n of V by sets V 1, V 2,..., V n. V i := {1, N} {i + 1, i + 2,..., N i} for sufficiently large N. All sets have the same maximal and minimal element. The second largest elements are decreasing and second smallest elements are increasing. V 1 V 2 V n Figure: Sketch of the sets V 1, V 2 and V n Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 18
39 The Total Max-Dominance-Strict-Independence Problem For the triple (a, b, c) R represented by the sets A, B, C we add the following sets, where k is unique for this triple: A \ {k}, B \ {k}, B \ {k + 1}, C \ {k + 1}, A \ {k + 2}, B \ {k + 2}, B \ {k + 3}, C \ {k + 3} We want B \ {k + 1} A \ {k}, B \ {k} C \ {k + 1}, A \ {k + 2} B \ {k + 3} and C \ {k + 3} B \ {k + 2} Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 19
40 The Total Max-Dominance-Strict-Independence Problem A \ {k} A \ {k + 2} B \ {k} B \ {k + 1} B \ {k + 2} B \ {k + 3} C \ {k + 1} A B C Figure: Family that forces that A B leads to B C C \ {k + 3} A B C Figure: Family that forces that A B leads to B C Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 20
41 The Total Max-Dominance-Strict-Independence Problem For the triple (a, b, c) R represented by the sets A, B, C we add the following sets, where k is unique for this triple: A \ {k}, B \ {k}, B \ {k + 1}, C \ {k + 1}, A \ {k + 2}, B \ {k + 2}, B \ {k + 3}, C \ {k + 3} We want B \ {k + 1} A \ {k}, B \ {k} C \ {k + 1}, A \ {k + 2} B \ {k + 3} and C \ {k + 3} B \ {k + 2} For example, we can force B \ {k + 1} A \ {k} by adding A \ {k, k + 4}, B \ {k + 1, k + 4} and either A \ {1, k, k + 4}, B \ {1, k + 1, k + 4} or A \ {k, k + 4, N}, B \ {k + 1, k + 4, N} Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 21
42 A summary of our results Not total Total Dom + Ind always NP-complete Max Dom +Ind always NP-complete Dom + Strict Ind in P NP-complete Max Dom + Strict Ind in P NP-complete Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 22
43 Future work The complexity of the studied problems if X is given in a compact way. Characterize the sets X that have orders satisfying (maximal) dominance and (strict) independence. Study other axioms and interpretations. Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 23
Ranking Specific Sets of Objects
Datenbank Spektrum (2017) 17:255 265 https://doi.org/10.1007/s13222-017-0264-7 SCHWERPUNTBEITRAG Ranking Specific Sets of Objects Jan Maly 1 Stefan Woltran 1 Received: 1 June 2017 / Accepted: 23 August
Preference Orders on Families of Sets When Can Impossibility Results Be Avoided?
Preference Orders on Families of Sets When Can Impossibility Results Be Avoided? Jan Maly 1, Miroslaw Truszczynski 2, Stefan Woltran 1 1 TU Wien, Austria 2 University of Kentucky, USA jmaly@dbai.tuwien.ac.at,
RANKING SETS OF OBJECTS
17 RANKING SETS OF OBJECTS Salvador Barberà* Walter Bossert** and Prasanta K. Pattanaik*** *Universitat Autònoma de Barcelona **Université de Montréal and C.R.D.E. ***University of California at Riverside
Well-Ordering Principle. Axiom: Every nonempty subset of Z + has a least element. That is, if S Z + and S, then S has a smallest element.
Well-Ordering Principle Axiom: Every nonempty subset of Z + has a least element. That is, if S Z + and S, then S has a smallest element. Well-Ordering Principle Example: Use well-ordering property to prove
Logic and Artificial Intelligence Lecture 22
Logic and Artificial Intelligence Lecture 22 Eric Pacuit Currently Visiting the Center for Formal Epistemology, CMU Center for Logic and Philosophy of Science Tilburg University ai.stanford.edu/ epacuit
a + b = b + a and a b = b a. (a + b) + c = a + (b + c) and (a b) c = a (b c). a (b + c) = a b + a c and (a + b) c = a c + b c.
Properties of the Integers The set of all integers is the set and the subset of Z given by Z = {, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, }, N = {0, 1, 2, 3, 4, }, is the set of nonnegative integers (also called
Properties of the Integers
Properties of the Integers The set of all integers is the set and the subset of Z given by Z = {, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, }, N = {0, 1, 2, 3, 4, }, is the set of nonnegative integers (also called
Ranking Sets of Objects
Cahier 2001-02 Ranking Sets of Objects BARBERÀ, Salvador BOSSERT, Walter PATTANAIK, Prasanta K. Département de sciences économiques Université de Montréal Faculté des arts et des sciences C.P. 6128, succursale
Axiomatic Decision Theory
Decision Theory Decision theory is about making choices It has a normative aspect what rational people should do... and a descriptive aspect what people do do Not surprisingly, it s been studied by economists,
Chapter 2 Axiomatic Set Theory
Chapter 2 Axiomatic Set Theory Ernst Zermelo (1871 1953) was the first to find an axiomatization of set theory, and it was later expanded by Abraham Fraenkel (1891 1965). 2.1 Zermelo Fraenkel Set Theory
PHIL 308S: Voting Theory and Fair Division
PHIL 308S: Voting Theory and Fair Division Lecture 12 Eric Pacuit Department of Philosophy University of Maryland, College Park ai.stanford.edu/ epacuit epacuit@umd.edu October 18, 2012 PHIL 308S: Voting
Ranking Sets of Objects by Using Game Theory
Ranking Sets of Objects by Using Game Theory Roberto Lucchetti Politecnico di Milano IV Workshop on Coverings, Selections and Games in Topology, Caserta, June 25 30, 2012 Summary Preferences over sets
1 Initial Notation and Definitions
Theory of Computation Pete Manolios Notes on induction Jan 21, 2016 In response to a request for more information on induction, I prepared these notes. Read them if you are interested, but this is not
Complete Induction and the Well- Ordering Principle
Complete Induction and the Well- Ordering Principle Complete Induction as a Rule of Inference In mathematical proofs, complete induction (PCI) is a rule of inference of the form P (a) P (a + 1) P (b) k
Lexicographic Refinements in the Context of Possibilistic Decision Theory
Lexicographic Refinements in the Context of Possibilistic Decision Theory Lluis Godo IIIA - CSIC 08193 Bellaterra, Spain godo@iiia.csic.es Adriana Zapico Universidad Nacional de Río Cuarto - CONICET 5800
Technical Results on Regular Preferences and Demand
Division of the Humanities and Social Sciences Technical Results on Regular Preferences and Demand KC Border Revised Fall 2011; Winter 2017 Preferences For the purposes of this note, a preference relation
Preference, Choice and Utility
Preference, Choice and Utility Eric Pacuit January 2, 205 Relations Suppose that X is a non-empty set. The set X X is the cross-product of X with itself. That is, it is the set of all pairs of elements
1.2 Posets and Zorn s Lemma
1.2 Posets and Zorn s Lemma A set Σ is called partially ordered or a poset with respect to a relation if (i) (reflexivity) ( σ Σ) σ σ ; (ii) (antisymmetry) ( σ,τ Σ) σ τ σ = σ = τ ; 66 (iii) (transitivity)
Discrete Optimization
Prof. Friedrich Eisenbrand Martin Niemeier Due Date: April 15, 2010 Discussions: March 25, April 01 Discrete Optimization Spring 2010 s 3 You can hand in written solutions for up to two of the exercises
Definitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations
Page 1 Definitions Tuesday, May 8, 2018 12:23 AM Notations " " means "equals, by definition" the set of all real numbers the set of integers Denote a function from a set to a set by Denote the image of
Committee Selection with a Weight Constraint Based on Lexicographic Rankings of Individuals
Committee Selection with a Weight Constraint Based on Lexicographic Rankings of Individuals Christian Klamler 1, Ulrich Pferschy 2, and Stefan Ruzika 3 1 University of Graz, Institute of Public Economics,
Regular Choice and the Weak Axiom of Stochastic Revealed Preference
Regular Choice and the Weak xiom of Stochastic Revealed Preference Indraneel Dasgupta School of Economics, University of Nottingham, UK. and Prasanta K. Pattanaik Department of Economics, University of
Utility Representation of Lower Separable Preferences
Utility Representation of Lower Separable Preferences Özgür Yılmaz June 2008 Abstract Topological separability is crucial for the utility representation of a complete preference relation. When preferences
Decisions under Uncertainty. Logic and Decision Making Unit 1
Decisions under Uncertainty Logic and Decision Making Unit 1 Topics De7inition Principles and rules Examples Axiomatisation Deeper understanding Uncertainty In an uncertain scenario, the decision maker
The definition of a vector space (V, +, )
The definition of a vector space (V, +, ) 1. For any u and v in V, u + v is also in V. 2. For any u and v in V, u + v = v + u. 3. For any u, v, w in V, u + ( v + w) = ( u + v) + w. 4. There is an element
7 Mathematical Induction. For the natural numbers N a the following are equivalent:
7 Mathematical Induction Definition: N a = {n N n a}. For the natural numbers N a the following are equivalent: 1. Ordinary axiom of induction: IfS N a such that (a) a S (b) n N a (n S n +1 S) then S =
Social Choice Theory for Logicians Lecture 5
Social Choice Theory for Logicians Lecture 5 Eric Pacuit Department of Philosophy University of Maryland, College Park ai.stanford.edu/ epacuit epacuit@umd.edu June 22, 2012 Eric Pacuit: The Logic Behind
Topics in Logic, Set Theory and Computability
Topics in Logic, Set Theory and Computability Homework Set #3 Due Friday 4/6 at 3pm (by email or in person at 08-3234) Exercises from Handouts 7-C-2 7-E-6 7-E-7(a) 8-A-4 8-A-9(a) 8-B-2 8-C-2(a,b,c) 8-D-4(a)
Modeling Preferences with Formal Concept Analysis
Modeling Preferences with Formal Concept Analysis Sergei Obiedkov Higher School of Economics, Moscow, Russia John likes strawberries more than apples. John likes raspberries more than pears. Can we generalize?
Relations. Binary Relation. Let A and B be sets. A (binary) relation from A to B is a subset of A B. Notation. Let R A B be a relation from A to B.
Relations Binary Relation Let A and B be sets. A (binary) relation from A to B is a subset of A B. Notation Let R A B be a relation from A to B. If (a, b) R, we write a R b. 1 Binary Relation Example:
Relations. Relations. Definition. Let A and B be sets.
Relations Relations. Definition. Let A and B be sets. A relation R from A to B is a subset R A B. If a A and b B, we write a R b if (a, b) R, and a /R b if (a, b) / R. A relation from A to A is called
Mathematical Social Sciences
Mathematical Social Sciences 74 (2015) 68 72 Contents lists available at ScienceDirect Mathematical Social Sciences journal homepage: www.elsevier.com/locate/econbase Continuity, completeness, betweenness
Rationality and solutions to nonconvex bargaining problems: rationalizability and Nash solutions 1
Rationality and solutions to nonconvex bargaining problems: rationalizability and Nash solutions 1 Yongsheng Xu Department of Economics Andrew Young School of Policy Studies Georgia State University, Atlanta,
A Generic Approach to Coalition Formation
A Generic Approach to Coalition Formation Krzysztof R. Apt and Andreas Witzel Abstract We propose an abstract approach to coalition formation by focusing on partial preference relations between partitions
A Characterization of Optimality Criteria for Decision Making under Complete Ignorance
Proceedings of the Twelfth International Conference on the Principles of Knowledge Representation and Reasoning (KR 2010) A Characterization of Optimality Criteria for Decision Making under Complete Ignorance
Individual decision-making under certainty
Individual decision-making under certainty Objects of inquiry Our study begins with individual decision-making under certainty Items of interest include: Feasible set Objective function (Feasible set R)
Technical R e p o r t. Merging in the Horn Fragment DBAI-TR Adrian Haret, Stefan Rümmele, Stefan Woltran. Artificial Intelligence
Technical R e p o r t Institut für Informationssysteme Abteilung Datenbanken und Artificial Intelligence Merging in the Horn Fragment DBAI-TR-2015-91 Adrian Haret, Stefan Rümmele, Stefan Woltran Institut
Static Decision Theory Under Certainty
Static Decision Theory Under Certainty Larry Blume September 22, 2010 1 basics A set of objects X An individual is asked to express preferences among the objects, or to make choices from subsets of X.
Modal and temporal logic
Modal and temporal logic N. Bezhanishvili I. Hodkinson C. Kupke Imperial College London 1 / 83 Overview Part II 1 Soundness and completeness. Canonical models. 3 lectures. 2 Finite model property. Filtrations.
QUASI-PREFERENCE: CHOICE ON PARTIALLY ORDERED SETS. Contents
QUASI-PREFERENCE: CHOICE ON PARTIALLY ORDERED SETS ZEFENG CHEN Abstract. A preference relation is a total order on a finite set and a quasipreference relation is a partial order. This paper first introduces
A General Impossibility Result on Strategy-Proof Social Choice Hyperfunctions
A General Impossibility Result on Strategy-Proof Social Choice Hyperfunctions Selçuk Özyurt and M. Remzi Sanver May 22, 2008 Abstract A social choice hyperfunction picks a non-empty set of alternatives
Week 4-5: Generating Permutations and Combinations
Week 4-5: Generating Permutations and Combinations February 27, 2017 1 Generating Permutations We have learned that there are n! permutations of {1, 2,...,n}. It is important in many instances to generate
Numerical representations of binary relations with thresholds: A brief survey 1
Numerical representations of binary relations with thresholds: A brief survey 1 Fuad Aleskerov Denis Bouyssou Bernard Monjardet 11 July 2006, Revised 8 January 2007 Typos corrected 1 March 2008 Additional
Ranking Sets of Possibly Interacting Objects Using Sharpley Extensions
Ranking Sets of Possibly Interacting Objects Using Sharpley Extensions Stefano Moretti, Alexis Tsoukiàs To cite this version: Stefano Moretti, Alexis Tsoukiàs. Ranking Sets of Possibly Interacting Objects
Supremum and Infimum
Supremum and Infimum UBC M0 Lecture Notes by Philip D. Loewen The Real Number System. Work hard to construct from the axioms a set R with special elements O and I, and a subset P R, and mappings A: R R
Talking Freedom of Choice Seriously
Talking Freedom of Choice Seriously Susumu Cato January 17, 2006 Abstract In actual life, we face the opportunity of many choices, from that opportunity set, we have to choose from different alternatives.
Chapter 1 - Preference and choice
http://selod.ensae.net/m1 Paris School of Economics (selod@ens.fr) September 27, 2007 Notations Consider an individual (agent) facing a choice set X. Definition (Choice set, "Consumption set") X is a set
OPERADS OF FINITE POSETS
OPERADS OF FINITE POSETS Abstract. We describe four natural operad structures on the vector space generated by isomorphism classes of finite posets. The three last ones are set-theoretical and can be seen
Algebraic Proof Systems
Algebraic Proof Systems Pavel Pudlák Mathematical Institute, Academy of Sciences, Prague and Charles University, Prague Fall School of Logic, Prague, 2009 2 Overview 1 a survey of proof systems 2 a lower
The natural numbers. Definition. Let X be any inductive set. We define the set of natural numbers as N = C(X).
The natural numbers As mentioned earlier in the course, the natural numbers can be constructed using the axioms of set theory. In this note we want to discuss the necessary details of this construction.
The Axiomatic Method in Social Choice Theory:
The Axiomatic Method in Social Choice Theory: Preference Aggregation, Judgment Aggregation, Graph Aggregation Ulle Endriss Institute for Logic, Language and Computation University of Amsterdam Ulle Endriss
Abstract model theory for extensions of modal logic
Abstract model theory for extensions of modal logic Balder ten Cate Stanford, May 13, 2008 Largely based on joint work with Johan van Benthem and Jouko Väänänen Balder ten Cate Abstract model theory for
Homework 3 Solutions, Math 55
Homework 3 Solutions, Math 55 1.8.4. There are three cases: that a is minimal, that b is minimal, and that c is minimal. If a is minimal, then a b and a c, so a min{b, c}, so then Also a b, so min{a, b}
NOTES (1) FOR MATH 375, FALL 2012
NOTES 1) FOR MATH 375, FALL 2012 1 Vector Spaces 11 Axioms Linear algebra grows out of the problem of solving simultaneous systems of linear equations such as 3x + 2y = 5, 111) x 3y = 9, or 2x + 3y z =
On the Complexity of the Reflected Logic of Proofs
On the Complexity of the Reflected Logic of Proofs Nikolai V. Krupski Department of Math. Logic and the Theory of Algorithms, Faculty of Mechanics and Mathematics, Moscow State University, Moscow 119899,
Coalitionally strategyproof functions depend only on the most-preferred alternatives.
Coalitionally strategyproof functions depend only on the most-preferred alternatives. H. Reiju Mihara reiju@ec.kagawa-u.ac.jp Economics, Kagawa University, Takamatsu, 760-8523, Japan April, 1999 [Social
PRACTICE PROBLEMS: SET 1
PRACTICE PROBLEMS: SET MATH 437/537: PROF. DRAGOS GHIOCA. Problems Problem. Let a, b N. Show that if gcd(a, b) = lcm[a, b], then a = b. Problem. Let n, k N with n. Prove that (n ) (n k ) if and only if
We want to show P (n) is true for all integers
Generalized Induction Proof: Let P (n) be the proposition 1 + 2 + 2 2 + + 2 n = 2 n+1 1. We want to show P (n) is true for all integers n 0. Generalized Induction Example: Use generalized induction to
LECTURE 3 LECTURE OUTLINE
LECTURE 3 LECTURE OUTLINE Differentiable Conve Functions Conve and A ne Hulls Caratheodory s Theorem Reading: Sections 1.1, 1.2 All figures are courtesy of Athena Scientific, and are used with permission.
Products, Relations and Functions
Products, Relations and Functions For a variety of reasons, in this course it will be useful to modify a few of the settheoretic preliminaries in the first chapter of Munkres. The discussion below explains
Graph Theory and Modal Logic
Osaka University of Economics and Law (OUEL) Aug. 5, 2013 BLAST 2013 at Chapman University Contents of this Talk Contents of this Talk 1. Graphs = Kripke frames. Contents of this Talk 1. Graphs = Kripke
06 Recursive Definition and Inductive Proof
CAS 701 Fall 2002 06 Recursive Definition and Inductive Proof Instructor: W. M. Farmer Revised: 30 November 2002 1 What is Recursion? Recursion is a method of defining a structure or operation in terms
Boolean Algebras. Chapter 2
Chapter 2 Boolean Algebras Let X be an arbitrary set and let P(X) be the class of all subsets of X (the power set of X). Three natural set-theoretic operations on P(X) are the binary operations of union
Outline. We will now investigate the structure of this important set.
The Reals Outline As we have seen, the set of real numbers, R, has cardinality c. This doesn't tell us very much about the reals, since there are many sets with this cardinality and cardinality doesn't
Relations, Functions, Binary Relations (Chapter 1, Sections 1.2, 1.3)
Relations, Functions, Binary Relations (Chapter 1, Sections 1.2, 1.3) CmSc 365 Theory of Computation 1. Relations Definition: Let A and B be two sets. A relation R from A to B is any set of ordered pairs
Nonmonotone Inductive Definitions
Nonmonotone Inductive Definitions Shane Steinert-Threlkeld March 15, 2012 Brief Review of Inductive Definitions Inductive Definitions as Sets of Clauses Definition A set B of clauses (of the form A b)
Non-Manipulable Domains for the Borda Count
Non-Manipulable Domains for the Borda Count Martin Barbie, Clemens Puppe * Department of Economics, University of Karlsruhe D 76128 Karlsruhe, Germany and Attila Tasnádi ** Department of Mathematics, Budapest
A Preference Logic With Four Kinds of Preferences
A Preference Logic With Four Kinds of Preferences Zhang Zhizheng and Xing Hancheng School of Computer Science and Engineering, Southeast University No.2 Sipailou, Nanjing, China {seu_zzz; xhc}@seu.edu.cn
Generating Permutations and Combinations
Generating Permutations and Combinations March 0, 005 Generating Permutations We have learned that there are n! permutations of {,,, n} It is important in many instances to generate a list of such permutations
Great Expectations. Part I: On the Customizability of Generalized Expected Utility*
Great Expectations. Part I: On the Customizability of Generalized Expected Utility* Francis C. Chu and Joseph Y. Halpern Department of Computer Science Cornell University Ithaca, NY 14853, U.S.A. Email:
Contents Propositional Logic: Proofs from Axioms and Inference Rules
Contents 1 Propositional Logic: Proofs from Axioms and Inference Rules... 1 1.1 Introduction... 1 1.1.1 An Example Demonstrating the Use of Logic in Real Life... 2 1.2 The Pure Propositional Calculus...
Mathematical Reasoning & Proofs
Mathematical Reasoning & Proofs MAT 1362 Fall 2018 Alistair Savage Department of Mathematics and Statistics University of Ottawa This work is licensed under a Creative Commons Attribution-ShareAlike 4.0
SYSU Lectures on the Theory of Aggregation Lecture 2: Binary Aggregation with Integrity Constraints
SYSU Lectures on the Theory of Aggregation Lecture 2: Binary Aggregation with Integrity Constraints Ulle Endriss Institute for Logic, Language and Computation University of Amsterdam [ http://www.illc.uva.nl/~ulle/sysu-2014/
A locally finite characterization of AE(0) and related classes of compacta
A locally finite characterization of AE(0) and related classes of compacta David Milovich Texas A&M International University david.milovich@tamiu.edu http://www.tamiu.edu/ dmilovich/ March 13, 2014 Spring
Decision Graphs - Influence Diagrams. Rudolf Kruse, Pascal Held Bayesian Networks 429
Decision Graphs - Influence Diagrams Rudolf Kruse, Pascal Held Bayesian Networks 429 Descriptive Decision Theory Descriptive Decision Theory tries to simulate human behavior in finding the right or best
Cowles Foundation for Research in Economics at Yale University
Cowles Foundation for Research in Economics at Yale University Cowles Foundation Discussion Paper No. 1904 Afriat from MaxMin John D. Geanakoplos August 2013 An author index to the working papers in the
General methods in proof theory for modal logic - Lecture 1
General methods in proof theory for modal logic - Lecture 1 Björn Lellmann and Revantha Ramanayake TU Wien Tutorial co-located with TABLEAUX 2017, FroCoS 2017 and ITP 2017 September 24, 2017. Brasilia.
m + q = p + n p + s = r + q m + q + p + s = p + n + r + q. (m + s) + (p + q) = (r + n) + (p + q) m + s = r + n.
9 The Basic idea 1 = { 0, 1, 1, 2, 2, 3,..., n, n + 1,...} 5 = { 0, 5, 1, 6, 2, 7,..., n, n + 5,...} Definition 9.1. Let be the binary relation on ω ω defined by m, n p, q iff m + q = p + n. Theorem 9.2.
Concepts for decision making under severe uncertainty with partial ordinal and partial cardinal preferences
Concepts for decision making under severe uncertainty with partial ordinal and partial cardinal preferences Christoph Jansen Georg Schollmeyer Thomas Augustin Department of Statistics, LMU Munich ISIPTA
Choosing the two finalists
Choosing the two finalists Economic Theory ISSN 0938-2259 Volume 46 Number 2 Econ Theory (2010) 46:211-219 DOI 10.1007/ s00199-009-0516-3 1 23 Your article is protected by copyright and all rights are
13 Social choice B = 2 X X. is the collection of all binary relations on X. R = { X X : is complete and transitive}
13 Social choice So far, all of our models involved a single decision maker. An important, perhaps the important, question for economics is whether the desires and wants of various agents can be rationally
On minimal models of the Region Connection Calculus
Fundamenta Informaticae 69 (2006) 1 20 1 IOS Press On minimal models of the Region Connection Calculus Lirong Xia State Key Laboratory of Intelligent Technology and Systems Department of Computer Science
Year 10 Unit G Revision Questions You can use a calculator on any question.
Year 10 Unit G Revision Questions You can use a calculator on any question. 1.) Find the mode, median, mean, range and interquartile range of each of the following lists. a.) 11, 13, 13, 16, 16, 17, 19,
Real Analysis Chapter 1 Solutions Jonathan Conder
3. (a) Let M be an infinite σ-algebra of subsets of some set X. There exists a countably infinite subcollection C M, and we may choose C to be closed under taking complements (adding in missing complements
Classes of Commutative Clean Rings
Classes of Commutative Clean Rings Wolf Iberkleid and Warren Wm. McGovern September 3, 2009 Abstract Let A be a commutative ring with identity and I an ideal of A. A is said to be I-clean if for every
10.4 The Kruskal Katona theorem
104 The Krusal Katona theorem 141 Example 1013 (Maximum weight traveling salesman problem We are given a complete directed graph with non-negative weights on edges, and we must find a maximum weight Hamiltonian
Linear Vector Spaces
CHAPTER 1 Linear Vector Spaces Definition 1.0.1. A linear vector space over a field F is a triple (V, +, ), where V is a set, + : V V V and : F V V are maps with the properties : (i) ( x, y V ), x + y
EQUIVALENCE RELATIONS (NOTES FOR STUDENTS) 1. RELATIONS
EQUIVALENCE RELATIONS (NOTES FOR STUDENTS) LIOR SILBERMAN Version 1.0 compiled September 9, 2015. 1.1. List of examples. 1. RELATIONS Equality of real numbers: for some x,y R we have x = y. For other pairs
Richter-Peleg multi-utility representations of preorders
Richter-Peleg multi-utility representations of preorders José Carlos R. Alcantud a, Gianni Bosi b,, Magalì Zuanon c a Facultad de Economía y Empresa and Multidisciplinary Institute of Enterprise (IME),
A proof of the Graph Semigroup Group Test in The Graph Menagerie
A proof of the Graph Semigroup Group Test in The Graph Menagerie by Gene Abrams and Jessica K. Sklar We present a proof of the Graph Semigroup Group Test. This Test plays a key role in the article The
Axioms for Set Theory
Axioms for Set Theory The following is a subset of the Zermelo-Fraenkel axioms for set theory. In this setting, all objects are sets which are denoted by letters, e.g. x, y, X, Y. Equality is logical identity:
Fuzzy Sets and Fuzzy Techniques. Sladoje. Introduction. Operations. Combinations. Aggregation Operations. An Application: Fuzzy Morphologies
Sets and Sets and Outline of Sets and Lecture 8 on Sets of 1 2 Centre for Image alysis Uppsala University 3 of February 15, 2007 4 5 Sets and Standard fuzzy operations Sets and Properties of the standard
HW 4 SOLUTIONS. , x + x x 1 ) 2
HW 4 SOLUTIONS The Way of Analysis p. 98: 1.) Suppose that A is open. Show that A minus a finite set is still open. This follows by induction as long as A minus one point x is still open. To see that A
Computational Social Choice: Spring 2015
Computational Social Choice: Spring 2015 Ulle Endriss Institute for Logic, Language and Computation University of Amsterdam Ulle Endriss 1 Plan for Today Last time we introduced the axiomatic method for
Non-zero-sum Game and Nash Equilibarium
Non-zero-sum Game and Nash Equilibarium Team nogg December 21, 2016 Overview Prisoner s Dilemma Prisoner s Dilemma: Alice Deny Alice Confess Bob Deny (-1,-1) (-9,0) Bob Confess (0,-9) (-6,-6) Prisoner
THE DIRECT SUM, UNION AND INTERSECTION OF POSET MATROIDS
SOOCHOW JOURNAL OF MATHEMATICS Volume 28, No. 4, pp. 347-355, October 2002 THE DIRECT SUM, UNION AND INTERSECTION OF POSET MATROIDS BY HUA MAO 1,2 AND SANYANG LIU 2 Abstract. This paper first shows how
Existence of Order-Preserving Functions for Nontotal Fuzzy Preference Relations under Decisiveness
axioms Article Existence of Order-Preserving Functions for Nontotal Fuzzy Preference elations under Decisiveness Paolo Bevilacqua 1 ID, Gianni Bosi 2, *, ID and Magalì Zuanon 3 1 Dipartimento di Ingegneria
Positively responsive collective choice rules and majority rule: a generalization of May s theorem to many alternatives
Positively responsive collective choice rules and majority rule: a generalization of May s theorem to many alternatives Sean Horan, Martin J. Osborne, and M. Remzi Sanver December 24, 2018 Abstract May
20 Ordinals. Definition A set α is an ordinal iff: (i) α is transitive; and. (ii) α is linearly ordered by. Example 20.2.
20 Definition 20.1. A set α is an ordinal iff: (i) α is transitive; and (ii) α is linearly ordered by. Example 20.2. (a) Each natural number n is an ordinal. (b) ω is an ordinal. (a) ω {ω} is an ordinal.